POW Homework

Question

Felicia would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Felicia cannot work more than a total of 20 hours per month. Let x represent the number of hours Felicia babysits and y represent the number of hours Felicia works at the ice cream shop.Create a system of linear inequalities and graph them below.



Which pairs (x,y) represent hours that Felicia could work to meet the given conditions. Circle all that apple

A.(4,15) B. (5,12) C. (10,9) D. (15,5) E. (19,1)

Given the conditions, if Felicia babysits for 7 hours this month what is the minimum number of hours she would have to work at the ice cream shop to earn at least $120 per month? Round your answer to the nearest whole hour.



Felicia prefers babysitting over working at the ice cream store. What is the maximum number of hours she can babysit to be able to earn at least $120 per month? Round your answer to the nearest whole hour.

Answer 1

Hi Kayjah,

 

Let's start by writing out the information they give us in the problem in the from of an equation:

x=babysitting hours paid at $5

y=ice cream shop hours paid at $8.

Total earned >120

total hours<20

So here is what our equations will look like:

5x+8y>120

x+y<20

 

Convert the first equation into y>mx+b format

8y>-5x+120

y>-5/8x+15

 

Convert the second equation into y>mx+b

x+y<20

y<-x+20

 

So the two equations we have now are:

y>-5/8x+15

y<-x+20

 

Graph the two lines y=-5/8x+15 and y=-x+20

 

Unfortunately you cannot paste a graph here, but here is a link to an online graphing calculator where I plotted the 2 lines.

 

http://www.meta-calculator.com/online/?panel-102-graph&data-bounds-xMin=-24.84375&data-bounds-xMax=25.15625&data-bounds-yMin=-28.82&data-bounds-yMax=29.740000000000002&data-equations-0=%22y%3D-5%2F8*x%2B15%22&data-equations-1=%22y%3D-x%2B20%22&data-points-0=%5B4%2C15%5D&data-points-1=%5B5%2C12%5D&data-points-2=%5B10%2C9%5D&data-points-3=%5B15%2C5%5D&data-points-4=%5B19%2C1%5D&data-rand=undefined&data-hideGrid=false

 

Since our equation is y>-5/8x+15, all points that fall on or above this line on the graph meet this requirement. (in red on the graph)

Our other equation is y<-x+20; so all points falling on or below this line meet the requirement. (in black on the graph)

 

Based on the graph, any points that fall above the red line on the graph and below the black line satisfy both equations, so just check each point listed in the question and see if it falls between the lines.

The points are in blue on the graph; you can see that (4,15), (5,12) and (10,9) are on or above the red and on or below the black, so they meet the conditions.  (15,5) and (19,1) are both on the black line but below the red, so they do not meet the conditions.

 

For the final part of the question:

What is the max number of hours she can babysit and still make at least $120?

In terms of our equations, this translates to what is the highest value for x that satisfies both equations?

 

In terms of the graph, we are looking for the point farthest to the right where the black line is not below the red line.  Basically, this means we are looking for the point where the 2 lines cross.  By looking at the graph, we can see that this happens at about (13.4,6.6)

We can also solve this mathmatically by finding the point where our equations are equal:

-5/8x+15=-x+20

Add x to both sides to get:

3/8x+15=20

subtract 15 from both sides:

3/8x=5

divide both sides by 3/8:

x=13.33

The question asks us to round to the nearest whole hour, so the minimum she can babysit is 14 hours to still make $120 a week.

 

Although I don't know why she'd rather babysit than work in an ice cream store.  

 

I know this is an insane amount of information to process.  Feel free to message me if you have any questions.

Thanks

Beth